YOURI PLOKHENKO* Research Center Planeta, ROSHYDROMET, Moscow, Russia W. PAUL MENZEL

Transcription

1 6 JOURNAL OF APPLIED ETEOROLOGY VOLUE 40 athematical Aspects in eteorological Processing of Infrared Spectral easurements from the GOES Sounder. Part I: Constructing the easurement Estimate Using Spatial Smoothing YOURI PLOKHENKO* Research Center Planeta, ROSHYDROET, oscow, Russia W. PAUL ENZEL Office of Research and Applications, NOAA/NESDIS, adison, Wisconsin (anuscript received January 000, in final form 6 July 000) ABSTRACT The accuracy of temperature and moisture vertical profiles retrieved from infrared spectral measurements is dependent on accurate definition of all contributions from the observed surface atmosphere system to the outgoing radiances. The associated inverse problem is ill posed. Instrument noise is a major contributor to errors in modeling spectral measurements. This paper considers an approach for noise reduction in the Geostationary Operational Environmental Satellite (GOES) spectral channels using spatial averaging that is based upon spectral characteristics of the measurements, spatial properties of atmospheric fields of temperature and moisture, and properties of the inverse problem. Spatial averaging over different fields of regard is studied for the GOES-8 sounder spectral bands. Results of the statistical analysis are presented.. Introduction Spatial filtering techniques are well known through their wide usage in reducing noise effects of different observing systems. The nominal objective is to determine a function (filter) over a spatially varying measurement that provides an optimal (in some sense) measurement estimate. Currently, in the routine processing of the Geostationary Operational Environmental Satellite (GOES) sounder spectral measurements, linear averaging over elements is used as the spatial filter for all spectral channels. This uniform spatial filter inadequately reduces the noise effects for some channels and overly smoothes the spatial variations for other channels. Our goal is to find an optimal size for the averaging area, also called the field of regard (FOR) in this paper, for each GOES-8 sounder spectral channel. In section, we consider the physical aspects of the noise filtering problem in atmospheric remote sensing. In section, a traditional mathematical model of noise filtering is introduced; then in section 4 we consider the results of a statistical spatial analysis of the spectral * Current affiliation: Cooperative Institute for eteorological Satellite Studies, adison, Wisconsin. Corresponding author address: Dr. Youri Plokhenko, CISS, University of Wisconsin adison, W. Dayton Street, adison, WI 706. measurements and estimate the appropriate size of the spatial filter for each GOES-8 sounder spectral channel. In section, results of temperature profile retrievals using spatially filtered spectral measurements are presented. Conclusions are offered in section 6.. Physical considerations GOES sounder measurements contain noise that varies substantially from one spectral channel to another. The measured radiances are a composite function of the optical properties of the GOES sounder instrument and the thermodynamical properties of the earth surface and atmosphere system. The latter are described by the surface emissivity, surface temperature, and atmospheric vertical profiles of moisture and temperature. These parameters have different spatial distributions (that can also be thought of as spectra). Surface properties of land can vary drastically and are found in the shortwave part of the spatial spectrum. Atmospheric moisture fields are smoother and correspondingly they are attributed to the median-wave part of the spatial spectrum; and atmospheric temperature fields are attributed in the first approximation to the longwave part of the spatial spectrum. Temperature and moisture fields become smoother with increasing height, and thus their spatial spectral properties shift in the direction of the longer spatial waves. We observe these spatial proper- 00 American eteorological Society

2 ARCH 00 PLOKHENKO AND ENZEL 7 ties in the GOES sounder spectral measurements of outgoing thermal radiances of the earth surface atmosphere system; a spectral measurement describes a specific atmospheric layer (sometimes in combination with the surface) and will have the corresponding spatial properties. Spectral bands sensitive to upper tropospheric and stratospheric layers, where radiances are not disturbed by clouds or the surface, exhibit large spatial uniformity and, therefore, shortwave spatial variation of the radiances describes the noise component of these measurements. The differences between spatial spectrums of measurement noise and meteorological parameters can be effectively used to reduce the influence of noise on the solution of the atmospheric profile retrieval; the objective is to estimate the spatial distribution of a meteorological parameter on the basis of measurements in the radiance spectrum. The physical model of a spectral measurement is based upon the radiative transfer equation (RTE), which is Fredholm s equation of the first kind. The corresponding inverse problem is ill posed. There is no one-to-one relation between spectral measurements and meteorological parameters. The solution of the problem is unstables; a small variation in a measurement and/or a physical model of a measurement can cause significant variation in the solution. It must be noted that the RTE is nonlinear with respect to all parameters: surface emissivity and temperature as well as atmospheric temperature and moisture profiles. For such an ill-posed problem, the mean solution of individual measurements (after spatial averaging of solutions within the field of regard) could substantially differ from the solution for the mean measurement (after spatial averaging of measurements within the field of regard). Thus, noise filtering will depend on the noise spectral distribution, the signal spectral and spatial distribution, the physical properties of the desired parameters and model parameters, and the numerical properties of the corresponding inverse problem.. Noise filtering We consider a formalism for noise filtering based on linear estimation. The measurements are described by the model f(x, y) f (x, y) (x, y), () where f (x, y) is a precise value of a function at a coordinate point (x, y), and f(x, y) is a measurement contaminated by a random noise (x, y). Our objective is to construct a function F[ f], which provides a measurement estimate fˆ F[ f] with the desired characteristics. When constructing F[ f], we assume that (x, y) is not correlated with f(x, y), and has a zero mean and spatial structure S(x, x, y, y) (x, y)(x, y) (x x)(y y), () where is noise variance and (x x) is the delta function defined by f(x) # f(x)(x x) dx, the symbol (...) designates the operation of averaging. This is the traditional white noise model. Furthermore, we consider f (x, y) to be differentiable. We define spatial averaging, F[ f], over the rectangular domain by the equation fˆ(x, y) F[ f ](x, y) () x y x y [f (x, y) (x, y)] dx dy, () where the unknown parameter defines the dimension of the spatial averaging. Using a first-order Taylor expansion of f (x, y) at(x, y) f (x, y) f (x, y) f (x, y) (x x) x f (x, y) (y y) R(x, y), (4) y where R(x, y) is the remainder term, we obtain ˆ fˆ(x, y) F[ f ](x, y) f (x, y) ˆ (x, y) R(x, y), () x y x y x y x y ˆ (x, y) () (x, y) dx dy, where ˆR(x, y) () R(x, y) dx dy. (6) The variance of the measurement estimate fˆ is defined by ( f ) ( f fˆ) ˆ R (ˆ ) R, (7) where f f fˆ and the symbol is now used from here on to designate parameter variation. For the white noise model expressed in (), the linear filter F[ f] will be optimal when ( f ) is minimized for ˆ (ˆ ) (8) 4 (here is dimensionless and is the corresponding conversion parameter with dimension km ). Supposing that f f f f max, (9) x y x y x,y ˆ over the averaging area and discarding the cross derivative f /x y, which is filtered out by the integration in (6), the term R(x, y) can be approximated by

3 8 JOURNAL OF APPLIED ETEOROLOGY VOLUE 40 f f R(x, y) max. (0) 6 x y x,y Last, we obtain the relation 4 ˆ ( f ) (ˆ ) R, () 4 9 which defines the optimal dimension ˆ for the spatial averaging 9 /6 ˆ. () 8 Equation () can be used for verifying a priori information about the statistical properties of the noise and for estimating the spatial properties of the measured spectral fields and the corresponding meteorological parameter fields. For that purpose, we have to determine [ f ()] for different values of. In practice we cannot measure [ f ()]. Only the measurement of f (x, y) is available, and the corresponding function [f ()] for f (x, y) is defined by (7) to be ˆ [ f ()] ( f fˆ) ˆ R 4. () 4 9 We can assess the applicability of the white noise model by inspecting whether or not () correctly describes [f()] as a function of. If we know from a priori meteorological information that the spatial variations of a parameter in a given atmospheric layer are small, so that K, (4) then we must obtain good correspondence between two measurements of [f ()] for close values and. If [f ( )] and [f ( )] satisfy () under condition (4), then the signal variation is negligible and the dimension of averaging can be increased to. In addition, () can be used for estimating the spatial variability of a field, if we know that K. In principle, we can use this approach {measuring [f()] for different } to estimate model components,, from which we can estimate the optimal ˆ from (). However, in practice, the measurement f(x, y) contains noise that is substantially different from the white noise model used. Thus all the relations introduced in the previous paragraphs would only represent approximations and can be used only as a reference in the data analysis. We need to solve () for the two unknowns and to establish a robust spatial analysis procedure; but () describes the signal properties and we need an estimate for describing the properties of the meteorological parameter field in (). To construct the required estimate, we suppose that the mean field f (x, y) from () is sufficiently smooth, so that it is three times differentiable with small variations of second derivatives and the negligibly small third derivatives f f f f K. () x y xy yx Then we can apply the Laplace operator L x y to the estimate fˆ(x, y) F[ f(x, y)]. The operator L is widely used in image processing (Hall 979; Rosenfeld and Kak 98). Using () and (), the differential of the resulting field can be expressed in the form [ ] [ ] ] () x ] () y fˆ fˆ L( fˆ) (x) (y) x y x,y f f x y x,y 4 (x) (y) () [ y (x) x x y L( f ) (y y) dy [ x (y) y y x L( f ) (x x) dx. (6) In (6) () is a function of noise, which can be written: () y y [(x x, y) (x x, y) (x x, y) (x x, y)] dy x [(x, y y ) (x, y y ) (x, y y ) (x, y y )] dx. x (7)

4 ARCH 00 PLOKHENKO AND ENZEL 9 For the noise model in (), () is characterized by () 0, () 4. The last two terms of (6), in accordance with (), can be discarded. Considering the spatial variations (x) (y) ( 4 ) for large, we obtain y (x) x L( f ) x (y y) dy () x y x (y) y L( f ) y (x x) dx 4 () y 6 x [ ] [ ] K. Thus the differential (6) has a mean [ ] f f 4 L( fˆ) x (x) y (y), (8) and a variance {L[ fˆ()]} () (9) It follows that L[fˆ()] describes the spatial properties of the signal model f (x, y), and {L[fˆ()]} describes properties of the noise model (x, y). Of course, (8) and (9), like (), will only be approximations; but the order of magnitude of their dependencies on will be important for us in analyzing () and (). Last, we define the corresponding relations for discrete rather than continuous measurements, f i,j. Given (n )/, where n is number of elements in one dimension of the square averaging domain, () is written [ ] (n ) 4 [ f ()], (0) (n ) 44 and (6) has form L( fˆ i,j) ( fˆ i,j fˆ i,j fˆ i,j fˆ i,j 4 fˆ i,j). () 4. Analysis of measurements GOES-8 sounder spectral channels and their intended purposes are presented in Table. The temperatureweighting functions, moisture sensitivities, and atmospheric transmittances are shown in Fig.. It follows from Fig. that there are many similarities in the vertical distribution of the contributions to the measurement among the different longwave and shortwave spectral bands: channels and (in Fig. a), channels 4,, and 4 (in Fig. a), and channels 6, 7,, and 6 (in Fig. b) have very similar weighting functions. From Figs. b,c, it follows that the measurements in channels 0 and will have similar spatial characteristics with respect to both temperature and moisture. Then the horizontal spatial structure of measurement variations in TABLE. Spectral channels of GOES-8 atmospheric sounder. Channel Central wavelength (m) Purpose: T temperature Q moisture Stratosphere T Tropopause T Upper-level T idlevel T Low-level T Surface T, Q Surface T, Q Surface T Total ozone Low-level Q idlevel Q Upper-level Q Low-level T idlevel T Upper-level T Boundary-layer T Surface T Surface T, Q channels 0 and with respect to the temperature should correspond to the horizontal spatial structure of measurements in spectral channels and. Instrument noise values (converted into a temperature value for a target temperature appropriate to each spectral channel observing a typical meteorological scene) are presented in Fig.. Significant variations from one spectral channel to another are evident. The statistical functions ( ) have been applied to measurements from 000 UTC on August 999; this nighttime scene was selected because the conditions on the surface were somewhat homogenous and solar reflection in the shortwave bands was avoided. All 8 channels were analyzed. We are especially interested in channels,,, 4,, 4, and because they are substantially affected by noise. They are not disturbed by surface effects (see Fig. ) and are sensitive to layers of the atmosphere where the temperature fields are sufficiently smooth. We expect the spatial smoothing to be most effective for noise reduction in these channels. The spatial characteristics of the measurements in channel are presented in Table. Here, N is the number of samples (in the picture fragment); N was a variable because only those locations (i, j) that satisfied the following spatial condition were included in the sample: fˆ ik,j fˆ i,j k,,0,. () Equation () was used to filter out obvious measurement errors and measurements in inhomogeneous scenes (e.g., partial cloudiness in the field of view). In constructing fˆ and calculating the associated statistics, we used measurements within the range 80 K f 40 K. Analogous to (), we introduced L( f i,j) ( fi,j fi,j fi,j fi,j 4 f i,j) () to describe the spatial variations in the original signal. From (), we have the noise model:

5 60 JOURNAL OF APPLIED ETEOROLOGY VOLUE 40 FIG.. Temperature weighting functions for spectral channels (a), 4, and and (b) 6 8, 0, 6, and 7. (c) easured brightness temperature changes in spectral channels 7 and 0 caused by a 0% perturbation of the atmospheric moisture profile. (d) Atmospheric (000 to 0 hpa) transmittances in channels 8. [L( f)] 4, (4) where is a space increment in a measurement matrix. Table contains results for the median and the linear estimate of the noise. For the median fˆ /, the error estimate ˆ / relates to the linear estimate error ˆ by (ˆ /) (ˆ ). () Considering [f()] in Table, we note that the median and linear estimates converge with increasing n, which implies from () and (0) that 4 /9 is negligibly small for the range of spatial averaging considered. Otherwise, we would observe a steady increase with increasing n. Thus, we can use for channel a field of regard of elements. It follows from TABLE. Spatial statistical characteristics of measurements from channel (GOES-8 sounder) for various fields of regard; there are (n ) fields of view in the field of regard. n N f (K) (f) (K ) L(fˆ) (K) [L(fˆ)] (K ) L ( f) (K) [L(f)] (K ) FIG.. Standard deviation of instrumental noise observed in measurements of the 8 spectral channels of GOES-8 sounder edian estimate Linear estimate

6 ARCH 00 PLOKHENKO AND ENZEL 6 FIG.. Channel variances (f) for different dimensions of the averaging domain: measured (solid line), and predicted from white noise model for two values of the noise variance (dashed and dotted lines). Table, that a linear estimate is preferable for averaging in channel, because the resulting field is smoother and more accurate. The linear estimate for elements maintains an approximation accuracy using [ˆ()] from (8) and {L[fˆ()]} from (9) defined by [ˆ()] {L[(fˆ()]} (6) that will be better than 0.0 K. Next we analyze the noise model from () using the appropriate value of [f()] from Table. The corresponding graphics are presented in Fig.. The measurements demonstrate substantially different properties than those predicted by the white noise model. This could be explained by the fact that the observed noise is as a mixture of two stochastic components,, (7) where component is the white noise, and component is described by another stochastic model differing from (). If and are not correlated, and does not correlate with the signal f(x, y), then we obtain for (7) by analogy with () and (0), ( f ) ( f fˆ) ˆ ˆ () { [ˆ (), ]} R, (8) where [ˆ(), ] designates some correlation function for the noise ˆ() in estimate () and the noise component in (7). In order for (8) to describe Fig., [ˆ(), ] must be a decreasing function of and. Both conditions are physically realizable and mathematically reasonable with respect to the basic model expressed in () (). Then from (8) it follows that the variance of observed noise (7) is described by the relation (f ) for large and 4 K. From Table it follows that in channel the noise variance has the level ( ).9 K. (9) We compare () from (9) with L( f) (last column in Table ), which can be expressed by analogy with (4) for noise model (7) as ( ) [ ( )] ( ) [L( f )]. K, (0) where () is a positive function 0 (), describing the spatial correlative properties of the noise component. We find (9) and (0) are not compatible for the estimates given in Table, if () 0. It indicates that the noise component is spatially correlated. Inequalities (9) and (0) confirm this. The estimate L[fˆ()] from Table along with (9) and (0) indicates that the noise model and corresponding estimate (8) do not describe properties of the observed noise; the model predicts an exponential decrease of L[fˆ()] with increasing ; in our experiment L[fˆ()] is close to a constant. The noise estimates from (9) and (0) are in good agreement with the data presented in Fig. ; estimates of () are less than the predicted value (4 K ) after filtering the noise by (). The filtering () also explains the number of samples N() associated with the increasing dimension of the field of regard: the larger the, the smoother the estimated field and correspondingly the fewer of estimates discarded. To summarize the results of the spatial analysis of the spectral measurements in GOES sounder channel, see the following: ) The measurements are very contaminated by noise, so that substantially different properties are observed than those described by a white noise model. The noise is spatially correlated. ) Variations in the thermal field are sufficiently small that a noise spatial filtering with an averaging domain of elements can be used. The comparison of median and linear estimates shows that the latter is preferable. The linear estimate indicates that the approximation accuracy variance is better than 0.06 K for a field of regard with elements. Estimates of spatial roughness [L(fˆ)] of averaged measurements for median and linear estimates in spectral channels of the GOES-8 sounder for different averaging domains (or fields of regard) are presented in Table. The spectral field is modeled using the temperature and moisture fields derived from the Eta odel forecast (Black 994; Rogers et al. 996) up to 00 hpa and extrapolated into the upper pressure levels hpa. The spatial roughness is less than 0.07 K in all spectral channels under a constant surface temperature and emissivity. The minimum spatial roughness of the averaged measurements should be found in the stratospheric channels and and upper-tropospheric chan-

7 6 JOURNAL OF APPLIED ETEOROLOGY VOLUE 40 TABLE. Spatial roughness of [L(fˆ)] (K ) of measurements in spectral channels of the GOES-8 sounder (median and linear estimations) for various fields of regard; there are (n ) fields of view in the field of regard. edian estimate Channel n n edian estimate Linear estimate nels,, and (see Fig. ), starting from 0.07 K. The spatial roughness should increase in the midtropospheric channels and reach a maximum of about K in channels 7, 8, 7, and 8 (the atmospheric window ). Using () to compare the spatial roughness for the median and linear estimations, we find that the linear estimate is optimal in channels,,, and. Figure 4 shows the spatial roughness [L(fˆ)] with spatial averaging of elements for all the spectral channels of the GOES-8 sounder (see Table, median estimate n ). The spectral distribution of spatial roughness does not correspond to the associated horizontal thermal structure of the atmosphere; consequently, the solution to the inverse problem will not reproduce them either. Comparing Figs. and 4 shows that spatial roughness from Fig. 4 reproduces spectral distribution of the noise maximums from Fig.. This means that a uniform spatial averaging is not effective. Figure contains the corresponding characteristics FIG. 4. Spatial roughness {L[ fˆ()]} of measurements after spatial averaging of elements (median estimate) in spectral channels of GOES-8 sounder. for measurements in channels 8. Statistics for channel are analogous to the statistics for channel. The quantity (f) slowly increases from.64 n to.96 n K. This implies that the spatial variation of the associated temperature field is small in comparison with the spatial averaging under consideration. We can use spatial averaging with a domain of elements ( ). The linear estimate suggests that {L[ fˆ ( )]} K ( f ) (ˆ ) {L[ fˆ ( )]} 0.04 K. Repeating the steps (7 0) for channel, we obtain ( ).6 K, and ( ) [ ( )] ( ) [L( f )]. K. To satisfy both inequalities, we again must consider that the observed noise in channel is spatially correlated. The noise estimates are in good agreement with data presented in Fig.. Here ( ) is slightly less than the predicted value ( K ) after the preliminary noise filtering by (). Statistics for channels and 4 demonstrate different properties than the statistics in channels and. We observe a linear increase of (f) with increasing that can be attributed to the signal spatial variation. Channel 8 atmospheric window measurements, which have a maximum surface contribution and therefore significant spatial variation, demonstrate a linear increase as well. Thus channels and 4 cannot be processed in the same way as channels and. Signal spatial variations [L( f)] for channels and 4 equal 4.0 and 4. K, respectively, which is orders of magnitude larger than the spatial variation obtained for channel (Fig. c). easurements in channels and 4 are not

8 ARCH 00 PLOKHENKO AND ENZEL 6 FIG.. (a) (b) Channel 8 variances (f ) and (c) (d) spatial roughness [L(fˆ)] for different dimensions of the averaging domain. contaminated by surface effects or atmospheric moisture. Channel must be spatially smoother than channel 4. In Figs. a,c, we find that the appropriate dimension for the spatial averaging is n for channel and n 4 for channel 4. Figure suggests an approximation accuracy of better than 0. K for channels and 4. easurements in channel 4 can be affected by clouds, thus a median estimate is preferable. Channel measurements are noticeably noisier (by a factor of at least ) than measurements in channel 4 (after averaging) and channel 6 (Fig. ). Channel measurements are affected by the surface and midtropospheric moisture (see Figs. c,d). Figures b,d statistics for channel, (f ) and [L(fˆ )], show features associated with window channels, in which measurement variations are mostly affected by spatial variations in surface temperature/emissivity and lower-tropospheric moisture. Channel should have substantially different spatial properties than an atmospheric window, but Fig. d shows that the atmospheric signal in channel has features dominated by the large surface signal in channel 8 (with respect to the spatial averaging). This contradiction suggests that the statistics in channels sensitive to the mid- and lower troposphere are affected by another factor. Another indication is the linear increase of [L(fˆ 6 8 )] with respect to (Fig. b). According to () and (0), [L(fˆ 6 8 )] should increase with 4 ; the discrepancy between measurement and prediction could be explained by an interaction of nonhomogeneity effects in the data sample between cloudy clear nonhomogeneity of atmospheric conditions, and land surface sea surface nonhomogeneity of surface conditions. In different conditions, the estimate of the second derivative in () and (0) will be substantially different. For example, in overcast conditions window channel measurements can be spatially smoother than atmospheric channel measurements. The same can be true for measurements in cloudless atmospheres over sea surfaces. Last, in channel we note that spatial averaging n avoids the spatial oscillation of moisture estimates in the midtroposphere (see Fig. c) and provides the same spatial smoothness as channel 4: [L(fˆ )] 0.4 K. In comparison, the natural signal spatial variation is [L( f )] 4.7 K. Channels 6 8 have significant signal spatial variation. The statistics [L( f 6 8 )] have values 6.0, 6., and 6.0 K, respectively. There is no physical reason to apply spatial averaging with these signal properties. Statistics for channels 9 6 are shown in Fig. 6. easurements in channel 9 depend on surface temperature and emissivity, upper-tropospheric temperature, and ozone. Channel 9 statistics are similar to those in channels 4 and in Figs. a,c, from which we conclude that n 9 should be applied. easurements in channels

9 64 JOURNAL OF APPLIED ETEOROLOGY VOLUE 40 FIG. 6. (a) (b) Channel 9 6 variances (f ) and (c) (d) spatial roughness [L(fˆ)] of estimates for different dimensions of averaging domain. 0 depend on mid- and upper-tropospheric temperature and moisture. The surface slightly affects measurements in channel 0. The statistics [L( f 0 )] have values.,.4, and 9. K, respectively. For comparison, in channels 7 and 8 the corresponding statistics have values 6. and 6.0 K, respectively. easurements in channels 0 and are smooth, in comparison with channels 8. Channel is noisy. Figure 6a shows that channels 0 and statistics have spatial variations. Figure 6c shows that spatial filtering in channels 0 and can cause a loss of information regarding moisture spatial variations. In Figure 6a, (f ) is slowly increasing with increasing n (or ). Channel reveals the spatial variations of midtropospheric temperature and upper-tropospheric moisture; both fields are sufficiently smooth. Corresponding thermal fields are described by measurements in channels and 4 (see Fig. a), for which we estimated the spatial variation to be K. From Fig. 6c, we see that the filter dimension n provides the required spatial smoothness in channel ; a linear estimate is preferable. In Figs. 6a,c, the observed noise in channel is noticeably larger, by a factor of at least 4, than the noise in Fig.. As with channels and, we conclude that the noise in channel has a spatially correlated component. When filtering channel, the scale of spatial variations in channels and must be similar (see Figs. b,c). To make them comparable in this sense, we use the filter with the dimension n in channel (see Fig. 6c). Last, we conclude from the statistics that channel 0 n 0 should be used to provide vertical homogeneous property to the inverse problem solution. Considering the statistical properties of measurements in channels 6 presented in Figs. 6b,d, we note that they complement the measurements in channels. Thus, the spatial features and accuracy approximations in channels 6 must have common scales with those in channels. The measurements in channels 4 and describe the mid- and uppertroposphere, respectively. They are analogous to channels and 4 in sounding the atmosphere (, 4 4 ). Channels and 6 are analogous to channels and 6. easurements in channels and 6 are affected by the surface (see Fig. d). Channels 6 are not affected by atmospheric moisture, unlike measurements in channels. Thus the measurement estimate in channels 6 should be spatially smoother than those in channels. Considering Figs. and 6, we notice the substantial differences between measurements in channels 6 and channels 4. The

10 ARCH 00 PLOKHENKO AND ENZEL 6 shape and amplitude of (f) and [L(fˆ)] in channels demonstrate that the measurements are very noisy (see also Fig. 4). Comparing channel in Figs. 6b,d with channels in Figs a,c, we find that channel noise has an analogous spatial structure. Filter estimate n in channel (see Fig. c) provides the spatial smoothness [L(fˆ )] 0.8 K. Figure 6d shows that we must use the filter dimensions n 4 and n 6 in channels 4 and to provide the required smoothness, defined by estimates in channels and 0. Channels and 6 have [L( f,6 )] of 9. and 7. K, respectively; these values are noticeably larger than [L( f,6 )] in the longwave channels and 6 (.0 and 6.0 K, respectively). We expect that [L( f,6 )] will have to satisfy conditions (see Fig. d and Fig. ): [L( f )] [L( f 6)] [L( f )] [L( f )] [L( f )],6 6, FIG. 7. (a) Channel 7 and 8 variances (f) and (b) spatial roughness [L(fˆ)] for different dimensions of averaging domain. Channel 8 also appears for comparison. that is, measurement properties in channels and 6 substantially differ from the model. Channels 7 and 8 have [L( f 7,8 )] of 9. and 7.8 K, respectively; these values are noticeably larger than [L( f 7,8 )] in the longwave window channels 7 and 8 (6. and 6.0 K, respectively). Here, (f) and [L(fˆ)] in channels 7 and 8 are presented in Fig. 7 (along with channel 8 for comparison). Figure 7a shows that (f) in channels 8, 7, and 8 is similar, indicating that there are some similarities in the measurement properties of the short- and longwave window channels. However, Fig. 7b shows that the measurements in the short- and longwave window channels have noticeably different spatial properties (see also Fig. 4). Those differences cannot be attributed to differences in noise. Here, (f) for channels 7 and 8 decreases faster than in channel 8; it becomes even less in channels 7 and 8 than in channel 8 with increasing n (or ). At larger n (or ) the difference seems to become constant. In contrast, channels 8 in Fig. d do not demonstrate such properties. Observed differences in [L( f 7,8 )], [L( f 7,8 )], [L(fˆ 7,8 )], and [L(fˆ 8)] could be explained by variations in surface optical properties within the spectrum. Thus properties (spatial and spectral) of the measurements in channels, 6 8 noticeably differ from the properties of the measurements in channels 8. The longwave spectral measurements play a basic role in the solution of inverse problem. The spatial smoothing is not applied to the measurements in channels 6 8, because the signal has a substantial spatial variation and the noise influence is comparatively small. However, the spatial filtering in channels, 6 8 is mostly responding to the properties of the inverse problem and the spectral model used, and therefore the consideration of the signal filtering is appropriate. Figures 6d and 7b (as compared with Fig. d) show that we can reduce observed differences between short- and longwave spectral measurements using the spatial filter with dimensions n, n 6, n 7, and n 8 in channels and 6 8. Table 4 presents a summary of the spatial averaging; it presents the dimensions of the spatial filter, an estimate of the spatial roughness [L(fˆ)] (K ), and the approximation accuracy (ˆ) (K ) for the 8 spectral channels of the GOES-8 sounder.. Comparing temperature retrievals for two spatial averaging strategies easurements from the GOES-8 sounder at 000 UTC on August 999 were processed using two spatial averaging strategies: the variable spatial averaging suggested in Table and the uniform spatial averaging currently used routinely (enzel et al. 998). Temperature profiles for the two datasets were compared with radiosonde observations (raob) from 00 UTC on August 999. The collocation distance is within 0. latitude and 0. longitude. Twenty-seven raobs and 8 retrievals (in cloud-free conditions) were matched. The geographical location of the collocated soundings is shown in Fig. 8a. The average absolute differences of GOES sounder temperature profiles versus raob temperature measurements for hpa are shown in

11 66 JOURNAL OF APPLIED ETEOROLOGY VOLUE 40 TABLE 4. Dimensions of spatial filter for noise reduction, spatial roughness, and approximation accuracy of measurement estimate in spectral channels of GOES-8 sounder (L linear, median). Channel Estimate [L(fˆ)](K ) ()(K ˆ ) Channel Estimate (L(fˆ)](K ) ()(K ˆ ) L L 0.04 L L L Fig. 8b. The results of processing radiances with the variable spatial averaging (plot ) are noticeably better than the results of processing radiances with the uniform spatial averaging (plot ). 6. Conclusions Noise reduction in the GOES sounder measurements was investigated in order to estimate the appropriate size of a simple square filter for each spectral channel as a function of the instrument noise and spatial variability of the detected radiation. A data analysis technique was developed using a traditional statistical approach to spatial filtering of noise. The physical basis for the technique is that estimates of the radiance fields (resulting from the spatial filtering) should be sufficiently smooth spatially and spectrally so that the retrieved meteorological fields exhibit appropriate spatial smoothness. In this sense, the technique can be considered as a simplification of D to D filtering, with some restrictions imposed on the spatial and spectral roughness of the spectral measurements. Analysis of GOES-8 sounder data indicates that the measurements are noisier than the instrument noise specification. In addition, the noise exhibits spatial correlation. Visual inspection of the spectral images reveals line to line spatially correlated noise attributed to detector recovery or smearing (after a hot scene the detector observes a cold scene ) or to calibration instability. For the GOES-8 sounder dataset from 000 UTC August 999, a spatial filter was estimated for each spectral channel. Profiles were retrieved from the radiative transfer equation. Solving the inverse problem involved estimating the surface emissivity and temperature (attributed to the shortwave spatial domain) as well as the atmospheric temperature and moisture profiles (where the spatial domain varies strongly with height from the shortwave in the atmospheric boundary layer to the longwave in the upper troposphere and stratosphere). The linear dimension of the spatial averaging filter ranges from element in the longwave spectral window (channels 6 8) to elements in the shortwave sounding channel. Retrievals from the variable filter are improved over those from a constant -element square filter. Acknowledgments. The authors gratefully acknowledge the useful discussions with r. Tim Schmit of the NESDIS Office of Research and Applications in the drafting of this paper. This work was supported by the NOAA/NESDIS Grant NA67EC000. FIG. 8. (a) Geographical location of GOES sounding points. (b) Average absolute difference of GOES sounder temperature retrievals vs raobs for spatial filtering with uniform dimensions inall spectral channels (plot ) and variable dimensions described in Table (plot ). REFERENCES Black, T. L., 994: The new NC esoscale Eta odel: Description and forecast examples. Wea. Forecasting, 9, Hall, E. L., 979: Computer Image Processing and Recognition. Academic Press, 84 pp. enzel, W. P., F. C. Holt, T. J. Schmit, R.. Aune, A. J. Schreiner,

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